For the equation \(x^2-2mx+3m=0\) to have no real roots, which is the correct interval for \(m\)?

Correct answer: \(0<m<3\)

A quadratic equation has no real roots when its discriminant satisfies \(D<0\). Here, \(a=1\), \(b=-2m\), and \(c=3m\), so \(D=b^2-4ac=4m^2-12m=4m(m-3)\). Thus, \(4m(m-3)<0\), which gives \(0<m<3\). At \(m=0\) or \(m=3\), \(D=0\), so the equation has real and equal roots rather than no real roots. Exam tip: remember that \(D<0\), \(D=0\), and \(D>0\) correspond respectively to no real roots, equal real roots, and two distinct real roots.